You are not cutting the engine until AFTER the spinning starts.. (You are correct though that once the engine IS cut, the spinning shouldn't get worse it should diminish slowly so I stand corrected there as far as this example goes.)
As far as autorotating normally, no you wouldn't because there would still be power being generated by the main rotor and while the opposite effect of spin would not be as drastic as it would be with the engine running, it would still be there and without a tail rotor you would still be spinning out of control... Albeit slightly slower.. Still uncontrolled nonetheless
(And this makes my head hurt! LOL)
DAMN the dynamics of helicopter flight is a daunting subject for sure.. Try reading the book by Wagtendonk "Principles of helicopter Flight"
http://www.barnesandnoble.com/w/principles-of-helicopter-flight-w-j-wagtendonk/1008248911?ean=9781560276494
That tomb will give you a headache in no time for sure!
LOL